Some extremal problems on the Riemannian sphere

In the paper, the open problem of the maximum product of the inner radii of $n$ domains in the case, when points and domains belong to the unit circle, is investigated. This problem is solved only for $n=2$ and $n=3$. No other results are known at present. We obtain the result for all $n \geqslant 2$. Also, we propose an approach that allows to establish evolutionary inequalities for the products of the inner radii of mutually non-overlapping domains.